Brief Summaries of Main Research Achievements
Two Different Descriptions:
I pressume the people in New York who ordered a total of 550 copies of my book on Quantum Mechanics only 4 months after publication probably realized its main issue. Namely that originating from my Ph.D. thesis (and thanks to my great supervisor Dingle) I solve, with an original and new perturbation method, 4 distinct types of Schrödinger equations (which people usually could not, or did not handle): a) for periodic potentials, b) for screened Coulomb potentials, c) for all quartic potentials, d) for the singular potential 1/r4 . The perturbation method is such that it allows the writing down of recurrence relations for the coecients of the perturbation expansions (no other perturbation method achieves this!), even solution of these recurrence relations and determination of the large order behaviour (and relation of this to level splitting and discontinuity across the energy cut), thus proving the eigenvalue expansions and solutions etc. are asymptotic expansions. In addition I present in the book the corresponding complimentary Feynman path integral calculations (originally done in collaboration with Liang). On top of these very dierent and original approaches, I show how agreement between the results of both methods is obtained. In the course of this work I realized that for all the fundamental potentials the small uctuation equation about the classical periodic instanton is a Lamé equation (other people were unfamiliar with this, I knew this from my Ph.D. work). Finally I considered the quantumclassical transition as a phase transition (originally suggested to me by the Russian Garanin). As one can see, there is a continuous path from my Ph.D. thesis to the end of my work. Until just before my retirement the singular potential problem was not complete. What gave me fuel and re to nish that, was my observation then, that this potential occurred in D3brane string theory, and some Americans had struggled with this.
Every student of physics knows the eigenenergies of the Schrödinger equation for Coulomb and harmonic oscillator potentials. But beyond that for physically important screened Coulomb potentials with resonances, anharmonic oscillators and periodic potentials? The derivation of real or complex eigenenergies for these (other than with rough approximations) required a new perturbation method, i.e. that of my Ph.D. supervisor Dingle and myself, and was achieved only stepwise, one case after another, over many years apart from the important early work, mostly by myself, including the solution of related problems, one being the complete solution of a Schrödinger equation with a singular potential. All this research was motivated by developments in theoretical particle physics as also the parallel path integral method which may be visualized as quantum mechanics about Newton's equation. A considerable part of my later research work was concerned with the rederivation of eigenenergies by this path integral method in which socalled periodic instantons play a role similar to that of solutions of Newton's equation. These two supplementary methods constituted the main thrust of my lifetime's research, along with mathematical byproducts such as asymptotic solutions of spheroidal and ellipsoidal wave equations, and further physics results in quantization of systems with constraints and the consideration of quantumthermal transitions as phase transitions (like that from ice to water).